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Small Worlds. Presented by Geetha Akula For the Faculty of Department of Computer Science, CALSTATE LA. On 8 th June 07. Structure of the Thesis. Introduction The Small World Phenomenon Applications to Routing Modeling Internet Social Networks Bibliography. The Small World Phenomena.
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Small Worlds Presented by Geetha Akula For the Faculty of Department of Computer Science, CALSTATE LA. On 8th June 07
Structure of the Thesis • Introduction • The Small World Phenomenon • Applications to Routing • Modeling Internet • Social Networks • Bibliography
The Small World Phenomena • Stanely Milgram’ s work on the small world is responsible for the standard believe that “everyone is connected by a chain of about six steps” • Their experiment “Send a packet from sets of randomly selected people to a stock broker in Boston”
Graphs Small World Graph • Most Large Scale Sparse Networks are found to be of the small world type e.g. ‘Internet’, ‘Electronic Circuits’, ‘Neurons’, ‘Human beings’ (Friendship Networks) • ‘Six Degrees of Separation’ (Strangers -- Sociological Concept) • Mathematically: In between ‘Regular Networks’ and ‘Random Networks’ • Regular Graphs • High characteristic path length • High degree of clustering • Random Graphs • Low characteristic path length • Low degree of clustering • Graphs of real life networks lie in between these two extremes. • A small world graph is any graph with a relatively small characteristic path length and a relatively large Clustering coefficient.
Small World models • Watts and Strogatz (1998) • Very small number of long range contacts needed to decrease path lengths without much reduction in cliquishness. • Long range contact picked uniformly at random (u.a.r) • Small world networks in 3 different areas esp. spread of infectious disease. • Probabilistic reach. No specific destinations. • Doesn’t require knowledge of paths and no active path selection.
Navigability Model by Kleinberg • Let the routing algorithm take place on the following network model • Start with a d-dimensional grid • Add random edges between vertices v and w with a probability of • Theorem: The routing algorithm will find ‘short‘ paths, if and only if α = d • ‘short‘ means paths with a length of O(log n) from any given source to any given target vertex • Another interesting aspect of Milgram's experiment is why people are able to find short paths (inverse αth-power distribution) • The idea behind the greedy alg. is that for any α < d there are too little random edges to make the paths short • For α > d there are too many random edges, and hence too many choices to which the message could be passed on • The message will make a (long) random walk through the network
Barabasi-Albert Model • Preferential attachment defines the probability for a vertex to get an edge to the new vertex • network has to be expanding, growing. • This precondition of growth is very important as the idea of emergence comes with it. It is constantly evolving and adapting. • The second is the condition of preferential attachment • that is, nodes (webpages) will wish to link themselves to hubs (websites) with the most connections.
Applications to Computer Networks • P2P overlay networks • Distributed hashing protocols • Security systems in mobile ad hoc networks • Hybrid sensor networks • Referral systems • Links between webpages. • Freenet. • The Internet. • Large Scale Ad-hoc Multicast
Applications:Hybrid Sensor Networks • Sharma & Mazumdar (2005) – • Adding of a few shortcut wires between wireless sensors. • Reduced energy dissipation per node as well as non-uniformity in expenditure. • Deterministic as well as probabilistic placement of wires. • Few wires unlike 1 long range contact per node in Kleinberg’s model. One in a cell / group of cells of sensors is wired. • Very good performance in static sink node case • with addition of Θ(nl(n)/log n) wires, average hop count reduced to Θ(1/√l(n)) and EDS to Θ(1/l(n)). • In dynamic case, with greedy routing, hop count cant be reduced below Ω(1/l(n)).
Links Between Webpages • A study looked at homepages and mailing lists at Stanford and MIT. • Looked at the contents, out-links, and in-links. • Tried to determine association network from the webpage links. • Assumptions of the study: • Links are bidirectional. • Easy to weed out links where users don’t know each other. L = 0.35 + 2.06 log N
Findings: • Average 2.5 links per person. • This leads to 1265 users (58%) connected at Stanford. 9.2 hops average path. • It was 1281 users (85.6%) connected at MIT. 6.4 hops average path. • High clustering coefficient of 0.22 and 0.21 greater than that of random networks. • Conclusion – we have a small world network.
The Internet • A study found that at the site level, the Internet has a small characteristic path length, and a large clustering coefficient orders larger than that of a random network. • Can exploit this property to build a smarter search engine. • Look for documents corresponding to search string. • Identify strongly connected component, find largest. • Calculate score (path length, clustering coefficient).
Many real networks are small-world networks Albert and Barabasi. REVIEWS OF MODERN PHYSICS, 74 2002 48-97
Map of Internet Internet Mapping Project: http://research.lumeta.com/ches/map/gallery/index.html
